Integrand size = 29, antiderivative size = 65 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {3 \csc (c+d x)}{a^3 d}-\frac {\csc ^2(c+d x)}{2 a^3 d}+\frac {4 \log (\sin (c+d x))}{a^3 d}-\frac {4 \log (1+\sin (c+d x))}{a^3 d} \] Output:
3*csc(d*x+c)/a^3/d-1/2*csc(d*x+c)^2/a^3/d+4*ln(sin(d*x+c))/a^3/d-4*ln(1+si n(d*x+c))/a^3/d
Time = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.75 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {6 \csc (c+d x)-\csc ^2(c+d x)+8 \log (\sin (c+d x))-8 \log (1+\sin (c+d x))}{2 a^3 d} \] Input:
Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]
Output:
(6*Csc[c + d*x] - Csc[c + d*x]^2 + 8*Log[Sin[c + d*x]] - 8*Log[1 + Sin[c + d*x]])/(2*a^3*d)
Time = 0.33 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3315, 27, 99, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^5}{\sin (c+d x)^3 (a \sin (c+d x)+a)^3}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle \frac {\int \frac {\csc ^3(c+d x) (a-a \sin (c+d x))^2}{\sin (c+d x) a+a}d(a \sin (c+d x))}{a^5 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\csc ^3(c+d x) (a-a \sin (c+d x))^2}{a^3 (\sin (c+d x) a+a)}d(a \sin (c+d x))}{a^2 d}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\int \left (\frac {\csc ^3(c+d x)}{a^2}-\frac {3 \csc ^2(c+d x)}{a^2}+\frac {4 \csc (c+d x)}{a^2}-\frac {4}{a (\sin (c+d x) a+a)}\right )d(a \sin (c+d x))}{a^2 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\csc ^2(c+d x)}{2 a}+\frac {3 \csc (c+d x)}{a}+\frac {4 \log (a \sin (c+d x))}{a}-\frac {4 \log (a \sin (c+d x)+a)}{a}}{a^2 d}\) |
Input:
Int[(Cos[c + d*x]^2*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]
Output:
((3*Csc[c + d*x])/a - Csc[c + d*x]^2/(2*a) + (4*Log[a*Sin[c + d*x]])/a - ( 4*Log[a + a*Sin[c + d*x]])/a)/(a^2*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
Time = 2.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {3}{\sin \left (d x +c \right )}+4 \ln \left (\sin \left (d x +c \right )\right )-4 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{3}}\) | \(49\) |
default | \(\frac {-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {3}{\sin \left (d x +c \right )}+4 \ln \left (\sin \left (d x +c \right )\right )-4 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{3}}\) | \(49\) |
risch | \(\frac {2 i \left (-i {\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{3 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{a^{3} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {8 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{3}}+\frac {4 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d \,a^{3}}\) | \(100\) |
Input:
int(cos(d*x+c)^2*cot(d*x+c)^3/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d/a^3*(-1/2/sin(d*x+c)^2+3/sin(d*x+c)+4*ln(sin(d*x+c))-4*ln(1+sin(d*x+c) ))
Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.17 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {8 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 8 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 6 \, \sin \left (d x + c\right ) + 1}{2 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )}} \] Input:
integrate(cos(d*x+c)^2*cot(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
1/2*(8*(cos(d*x + c)^2 - 1)*log(1/2*sin(d*x + c)) - 8*(cos(d*x + c)^2 - 1) *log(sin(d*x + c) + 1) - 6*sin(d*x + c) + 1)/(a^3*d*cos(d*x + c)^2 - a^3*d )
\[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\cos ^{2}{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(cos(d*x+c)**2*cot(d*x+c)**3/(a+a*sin(d*x+c))**3,x)
Output:
Integral(cos(c + d*x)**2*cot(c + d*x)**3/(sin(c + d*x)**3 + 3*sin(c + d*x) **2 + 3*sin(c + d*x) + 1), x)/a**3
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {8 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} - \frac {8 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}} - \frac {6 \, \sin \left (d x + c\right ) - 1}{a^{3} \sin \left (d x + c\right )^{2}}}{2 \, d} \] Input:
integrate(cos(d*x+c)^2*cot(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
-1/2*(8*log(sin(d*x + c) + 1)/a^3 - 8*log(sin(d*x + c))/a^3 - (6*sin(d*x + c) - 1)/(a^3*sin(d*x + c)^2))/d
Time = 0.14 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {4 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} d} + \frac {4 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3} d} + \frac {6 \, \sin \left (d x + c\right ) - 1}{2 \, a^{3} d \sin \left (d x + c\right )^{2}} \] Input:
integrate(cos(d*x+c)^2*cot(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
-4*log(abs(sin(d*x + c) + 1))/(a^3*d) + 4*log(abs(sin(d*x + c)))/(a^3*d) + 1/2*(6*sin(d*x + c) - 1)/(a^3*d*sin(d*x + c)^2)
Time = 33.37 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.65 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^3\,d}-\frac {8\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^3\,d}+\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{2}\right )}{4\,a^3\,d} \] Input:
int((cos(c + d*x)^2*cot(c + d*x)^3)/(a + a*sin(c + d*x))^3,x)
Output:
(4*log(tan(c/2 + (d*x)/2)))/(a^3*d) - tan(c/2 + (d*x)/2)^2/(8*a^3*d) - (8* log(tan(c/2 + (d*x)/2) + 1))/(a^3*d) + (3*tan(c/2 + (d*x)/2))/(2*a^3*d) + (cot(c/2 + (d*x)/2)^2*(6*tan(c/2 + (d*x)/2) - 1/2))/(4*a^3*d)
Time = 0.19 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.17 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-32 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2}+16 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2}+\sin \left (d x +c \right )^{2}+12 \sin \left (d x +c \right )-2}{4 \sin \left (d x +c \right )^{2} a^{3} d} \] Input:
int(cos(d*x+c)^2*cot(d*x+c)^3/(a+a*sin(d*x+c))^3,x)
Output:
( - 32*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2 + 16*log(tan((c + d*x)/2) )*sin(c + d*x)**2 + sin(c + d*x)**2 + 12*sin(c + d*x) - 2)/(4*sin(c + d*x) **2*a**3*d)